Why do multiple copies appear in the spectrum of a sampled signal?

• A. The spectrum is convolved with a rectangular window in time, producing copies.
• B. The spectrum is unaffected by sampling.
• C. The spectrum is replicated at multiples of Fs due to sampling, with equal amplitude.
• D. The spectrum is inverted in frequency.

Answer: (C)

Sampling a signal in time causes its spectrum to repeat at intervals of the sampling frequency. This happens because sampling is multiplication of the signal by a periodic impulse train in time, and multiplication in time corresponds to convolution in frequency with that train’s spectrum. The spectrum of the impulse train is another train of impulses spaced by Fs, so the original spectrum is copied and shifted to every k·Fs. Each replica has the same shape as the original (and, under typical conventions, the same amplitude), just located at multiples of Fs. This is why you see multiple copies in the spectrum of a sampled signal. If the sampling rate is high enough that these copies don’t overlap, you can perfectly reconstruct the original signal; if not, they overlap and cause aliasing.